We begin this section with a remarkable identity due to Euler:
ei9 =cosO+ isinB, (5.3.1)
where we will choose 8 to be real. To prove this identity, we must define what we mean by e raised to a complex power i8. We define eanything to be the infinite power series associated with the exponential function e:r: with x replaced by anything. Thus
e•lephant = ~ (elephant)"
LJ n!
n=O
(5.3.2)
y
Figure 5.1. The complex plane.
which converges for any finite sized elephant.
Turning to our problem, we expand the infinite series for the exponential and collect the real and imaginary parts as follows:
oo (i9)n ei8 = """ L..J -;;:r
n=O
oo (-l)n(9)2n +. oo (-1)n(9)2n+l
= ~ (2n)! '~ (2n + 1)!
= cos9+isin9
(5.3.3) (5.3.4) (5.3.5) where we have used the fact that i2 = -1, i3 = -i, i4 = 1, and so on, as well as the infinite series that define the sine and cosine functions. (These expansions converge for all finite 9, as shown before. The presence of i does not in any way complicate the question of convergence since it either turns into a ±1 or into ±i.) Setting 9 = 1r we obtain one of the most remarkable formulae in mathematics:
(5.3.6) Who would have thought that 1r which enters as the ratio of circumference to diameter, e, as the natural base for logarithms, i, as the fundamental imaginary unit and 0 and 1 (which we know all about from infancy) would all be tied together in any way, not to mention such a simple and compact way? I hope I never stumble into anything like this formula, for nothing I do after that in life would have any significance.
Look at Fig. 5.1 of the complex plane and note that
Z = X +iy (5.3.7)
= r[;+i;] (5.3.8)
= r [cos8 + isin8] (5.3.9)
= rei9. (5.3.10)
The last equation is called the polar form of the complex number as compared to the cartesian form we have been using so far. One refers to 8 as the argument or phase of the number and r as its modulus or absolute value. It is just as easy to visualize the number in the complex plane given the polar form as it was with the cartesian form. (As is true with polar coordinates in any context, 8 is defined only modulo 211", that is to say adding 211" to it changes nothing. We can usually restrict it to the interval [0- 21r].) Att manipulations we did before in the cartesian form can of course be carried out in polar form, though some become easier and some harder. Thus if
z = re'9 then (5.3.11)
z* = re-i9 (5.3.12)
zz* = r2 (lzl=r) (5.3.13)
1 z = -e 1 -i9 (5.3.14)
r
ZtZ2 = TJT2ei(9t+82), (5.3.15) (We are using the fact that the law of composition of exponents under a product works for complex exponents as wett. Indeed this is built into the exponential function defined by the infinite series. You may check that this works to any given order even for imaginary arguments. In Chapter 6, this will be proven more directly.) The last formula tells us how easy it is to multiply or divide two complex numbers in polar form:
To multiply two complex numbers, multiply their moduli and add their phases.
To divide, divide by the modulus and subtract the phase of the denominator.
On the other hand to add two complex numbers we have to go back to the cartesian form, add the components and revert to the polar form.
Let us return to Eqn. (5.2.21) and manipulate the numbers in polar form. First
(v'2 -1} + i(v'2 + 1)
ZJ = 2 (5.3.16)
(v'2-1)2+(v'2+1)2 [ã t [l+v'2]]
= 4 exp urc an J2 _ 1 (5.3.17)
(5.3.18)
As for z2,
z2 = J~ + ~eiarctanl = eiTr/4 = e.785i.
Now it is easy to form the product and quotient
=
~. 1e(1.400+.785)i
~e 2 ã 185 i = 1.224 (cos2.185+isin2.185)
= --+i 1
V2
=
~e(1. 400 -ã 785 li = 1.224 (cos .615 + i sin .615) 1+-i
V2
in agreement with the calculation done earlier in cartesian form.
(5.3.19)
(5.3.20) (5.3.21) (5.3.22)
(5.3.23)
Complex numbers z = rei9 with r = 1 have lzl = 1 and are called unimodular.
We may imagine them as lying on a circle of unit radius in the complex plane.
Special points on this circle are
(J = 0 (1) (5.3.24)
7r/2 (i) (5.3.25)
1T (-1) (5.3.26)
= -1T /2 ( -i). (5.3.27)
You are expected to know these points at all times.
Problem 5.3.1. VerifY the correctness of the above using Euler's formula.
When we work with real numbers, we know that multiplication by a number, say 4, rescales the given number by 4. Multiplying a number in the complex plane by rei8 , rescales its length (or modulus) by r and also rotates it counterclockwise by 9. Multiplying by a unimodular number simply rotates without any rescaling.
Problem 5.3.2. For the following pairs of numbers, give their polar form, their complex conjugates, their moduli, product, the quotient zl! z2 , and the complex conjugate of the quotient:
1 + i
Zt = V2 Z2 = Va - i
3 + 4i [ 1 + 2i ] 2
Zl = 3 - 4i z2 = 1 - 3i
Problem S.3.3. Express the sum of the following in polar fonn:
Recall from Euler's fonnula that for real 8
cos8 . eiB + e-iB
= Re e'6 = -~--
2 (5.3.28)
. eiB _ e-iB
= Im e'6 = ã
sin8 2i (5.3.29)
You should remember the above results at all times.
Problem S.3.4. Check the following familiar trigonometric identities by expressing all functions in terms of exponentials: sin2 x + cos2 x = 1, sin 2x = 2 sin x cos x, cos 2x = cos2 x-sin2 x.
Problem S.3.S. Consider the series
(5.3.30) Sum this geometric series, take the real and imaginary parts of both sides and show that
sin2n8 cos8 + cos38 + ã ã ã + cos(2n- 1)8 = -.-2sm8
and that a similar sum with sines adds up to sin 2 n 8/ sin IJ.
Problem 5.3.6. Consider De Moivre's Theorem, which states that (cos8 +
i sin e)n = cos n8 + i sin n8. This follows from taking the n-th power of both sides of Euler's theorem. Find the formula for cos 41J and sin 48 is terms of cos 8 and sine. Given eiAei8 = ei(A+B) deduce cos(A +B) and sin(A +B).